In the Laboratory

JChemEd.chem.wisc.edu Vol. 76 No. 9 September 1999 Journal of Chemical Education 1269

Experiments involving the spectrum of atomic hydrogenare very useful for introducing the quantization of energy levels(13) to chemistry students. Using the Balmer series (4, 5), itturns out that the energy of the lone orbiting electron dependsexclusively on the principal quantum number, n. In this casethe electron is solely subject to the electrostatic influence ofthe nucleus. The sodium atom, as any alkali metal, resemblesthe hydrogen atom in having only one electron in the valenceshell. However, the interelectronic interactions must beconsidered in the energy calculations because of the penetrationof this electron into the inner closed shells. As a consequenceof the radial probability-density distribution (Fig. 1) the energylevels depend not just on the principal quantum number,but also on the angular momentum quantum number, l. Aquantity called the quantum defect, , which accounts forthe distinct capability of the valence electrons to penetrateinto the atoms inner closed shells, is usually introduced (5).1

On the other hand, many concepts on atomic structure(energy levels, ionization potentials, electronegativity of theelements, and so on) are discussed in terms of effective nuclearcharges. In the case of the valence electron of an alkali metalthis quantity involves the shielding effect of the inner-shellelectrons on the nuclear charge Z * (610). The effectivenuclear charge depends on n and l quantum numbers, andcan be used to describe the spectral series. It seems to be aneasier concept for students to understand than the quantumdefect, because it retains the meaning of an integer quantumnumber, as learned in basic courses. These advantages in usingZ * instead of , together with the method presented in thisarticle, overcome the advantage that is practically constantfor a given l (only one value of has to be calculated foreach angular momentum, from the experimental data).

We describe a simple graphical method for using spectraldata to determine the effective nuclear charges, Z *nl , feltby the valence electron of sodium in the ground and variousexcited energy states, the shielding constants nl , and theenergy levels. The shielding constant represents the amountof nuclear charge screened by the inner electrons on the outer

nl electron, nl = Z Z *nl , where Z is the total nuclear charge.This procedure has been used in our undergraduate physicalchemistry laboratory classes. Its accuracy is high enough fordidactic purposes.

This experiment is very convenient for introducing theconcept of angular momentum and its effect on the energiesof the orbitals, as well as the concepts of a main subject likespectroscopy.

Methods

The energy term for the sodium atom, Tnl = R(1/(n l )2),corresponds to the binding energy (cm{1) of an electron inan nl orbital. Introducing the effective nuclear charge theexpression becomes Tnl = R(Z *nl )2/n2. Therefore the Rydbergseries (4 ) for the sodium atom may be rewritten as:

Sharp series

s = RZ3p*

2

32

Zns*2

n2 n = 4, 5, 6, (1)

Principal series

p = RZ3s*

2

32

Znp*2

n2 n = 3, 4, 5, (2)

Diffuse series

d = RZ3p*

2

32

Znd*2

n2 n = 3, 4, 5, (3)

where l are the wavenumbers (cm{1) corresponding to thetransitions between the two terms in the second member.

The effective nuclear charge is a measure of the averagenuclear charge felt by the outermost electron in the variousorbitals, considering the interelectronic repulsions and itspenetration capability. Therefore, effective nuclear charges canbe evaluated from experimental data using the eqs 1 to 3.Instead of solving these equations analytically, a graphicalapproach using spreadsheet software (Origin, Excel, etc.) wasemployed.

Experimental Procedure

The sodium spectrum can be obtained with any kind ofspectrometer, from an old photographic instrument to amodern recording spectrophotometer. The spectrum shownin Figure 2 was obtained with a Jobin Yvon U-1000 instrumentthat gives the spectrum directly in cm{1. Owing to the expo-nential variation of the line intensities in a spectral series,the spectrum in the high-energy side was plotted on a loga-rithmic scale. In the 16,00018,500 cm{1 region the heightof the lines is the intensity (nonlogarithmic) after passing

A Procedure to Obtain the Effective Nuclear Chargefrom the Atomic Spectrum of Sodium

O. Sala,* K. Araki, and L. K. NodaInstituto de Qumica, Universidade de So Paulo, C.P. 26077, 05599-970, So Paulo, Brazil; *oswsala@quim.iq.usp.br

Figure 1. Radial probability-density distribution (distance in atomicunits) for the 3s, 3p, and 3d electrons in the sodium atom. Theshaded area corresponds to the core.

In the Laboratory

1270 Journal of Chemical Education Vol. 76 No. 9 September 1999 JChemEd.chem.wisc.edu

through a Corning glass filter #512 (or a neodymium saltsolution); this filter reduces drastically the intensity of theyellow line, indicated by (p). The spectrum can be obtainedin about half an hour. The calculations and interpretation ofthe results may take about four hours. The baseline in thehigh-frequency side shows a structure due to rotationalvibrational bands of Na2 molecules. This will not be discussedbecause it is beyond the scope of this article.

Identification of the Spectral Series

The first step in analyzing the emission spectrum is tocorrectly attribute the spectral series. By a close inspection ofthe spectrum (Fig. 2), disregarding the fine structure (onlythe transitions involving the 3P3/2 level, in the doublets, will beconsidered in the following), it is possible to distinguish twoprogressions of lines, indicated by (a) and (b), which becomecloser and closer going to higher wavenumbers. Consideringeqs 1 to 3, it is evident that when n the sharp and diffuseseries tend to the same limit, T3p = R(Z *3p )2/32. Therefore,the two progressions can be attributed to these series.

Assignment of Diffuse Series

The equation for the diffuse series, eq 3, can be writtenas d = A BX, where X = 1/n2, B = R(Z *nd )2. From a plot ofthe wavenumbers (cm{1) as a function of 1/n2 for the (a) and(b) progressions only the diffuse series should give a straightline because the d electron is almost nonpenetrating and Z *ndcan be assumed as 1. The correct attribution of the principalquantum numbers to the lines can be determined by trialand error, assigning different n values to the first line in thespectrum (starting with n = 3) until the best correlationcoefficient is obtained from a linear regression. In this way,the diffuse series can be identified as the progression (a) inFigure 2 and the Rydberg constant and the value of T3p can beobtained, respectively, from the slope (B) and the linear co-efficient (A) of this straight line. We obtained R = 110,562 cm{1

(lit. 109,734 cm{1 [5]) and T3p = 24,489 cm{1 (lit. 24,476 cm{1

[11]), as shown in Figure 3.

Assignment of Sharp Series and Calculation of Z *nsThe values of (Z *ns )2 for the ns levels can be determined

by substituting the appropriate s (beginning with the first ofthe (b) lines in Fig. 2) and n values into the expression (Z *ns )2 =(T3p s)n2/R (T3p = 24,489 cm{1 and R = 110,562 cm{1 werepreviously determined).

The correct attribution of n to the s (with n 4) canbe obtained calculating the energy terms (Tns = R(Z *ns )2/n2)from the (Z *ns )2 above, for n = 4, 5, 6, , for n = 5, 6, 7, and for n = 6, 7, 8, . It is observed that only the set ofvalues starting with n = 5 gives the correct values of the energyterms by comparison with the values from the literature (11).

It is useful to find an expression correlating (Z *ns )2 withany value of the principal quantum number, because it allowsthe prediction of the spectral lines even for those transitionsout of the observed spectral region. As the effective nuclearcharge is expected to decrease with increasing n for a fixed l,plots of (Z *ns )2 versus 1/n and 1/n2 were tried. The plot as afunction of 1/n2 (Fig. 4(a)) shows a linear dependence, fittedby the linear regression

(Z *ns)2 = 17.695/n2 + 1.162 (4)

Figure 3. Plot of experimental wavenumbers for the diffuse seriesas a function of 1/n2 (filled squares) and the corresponding linearregression (dotted line).

Figure 4. (a): Plot of (Z*ns)2 as a function of 1/n2 with the (Z*ns)2calculated from the spectral values of the sharp series (filled squares)and the corresponding linear regression (full line; eq 2). (b): Plotof (Z*np)2 as a function of 1/n2 for the principal series (filled squares)and the corresponding linear regression (full line; eq 3).

Figure 2. Emission spectrum of the sodium atom, intensity in arbitraryunits. In the 16,00018,500 cm{1 region a didymium No. 512filter (Corning) was employed to decrease the intensity of the yellowline (p). In the high-energy side, a logarithmic scale was usedbecause of the enormous difference in the intensities of the lines.

In the Laboratory

JChemEd.chem.wisc.edu Vol. 76 No. 9 September 1999 Journal of Chemical Education 1271

From this equation the squared effective nuclear chargecan be estimated for any value of n 4 (see Table 1). The T3sand (Z *3s )2 values can be determined straightforwardly fromthe first line of the principal series (yellow line); since T3p isalready known and T3s T3p = 16,978 cm{1, we have (Z *3s )2 =41,467 32/R.

Calculation of Z *np and the Energy Terms

Unfortunately, only the 3S3P transition of the principalseries is observed in the visible region of the spectrum (indi-cated by (p) in Fig. 2). The remaining transitions have lines inthe ultraviolet region and are filtered by the glass tube of thesodium lamp. Even so, it is possible to find the relationship forthe (Z *np )2 because as 1/n2 tends to zero, (Z *np )2 and (Z *ns )2 con-verge to the same limit; that is, when n , (Z *

p)2 =(Z *

s)2 = 1.162, the linear coefficient of eq 4. The value of (Z *3p )2

can be calculated from the term T3p, (Z *3p )2 = 24,488 32/R =1.9935. These two points define a straight line (Fig. 4(b))described by the equation

(Z *np )2 = 7.482/n2 + 1.162 (5)

This equation can be used to evaluate the squared effectivenuclear charges of the np electrons for n 3. From these valuesit is possible to calculate the energies of the transitions inthe ultraviolet region. The validity of eq 5 can be verified bycomparison with the energy levels from the literature (11).

The calculated effective charges and their squared valuesare listed in Table 1, showing their dependence on the angularmomentum and principal quantum numbers. As expected,the values of the calculated effective nuclear charges followthe sequence

Z *ns > Z *np > Z *ndTo verify the reliability of this new procedure, the energies

of the ns, np, and nd electrons (3 n 10) were calculated,according to Tnl = R(Z *nl )2/n2, and compared with the valuesfrom the literature (Table 2). The errors are reasonably lowand of the same magnitude as those obtained using themethod of quantum defects (12).

The shielding factor nl can be calculated from the ex-pression (Z nl ) = Z *nl (Z = atomic number). Its values arepresented in Table 1 for the ns and np levels. Asexpected, nl is a function of the quantum num-bers n and l, because the ability of the valence elec-tron to penetrate into the closed shells depends onthe level in which it is found.

In conclusion, our method makes it possible todetermine experimentally the effective nuclearcharges, the shielding factors, and the energiesof the several states of the valence electron of thesodium atom with good accuracy. The most im-portant point is that this charge can be consideredas responsible for the effect of the interelectronicinteractions without changing the integer value ofthe quantum number in the denominator in theRydberg equations.

Note1. The quantum defect is a function of the angular momentum

quantum number and is practically independent of the principalquantum number. Using the Rydberg series (12, 13) the quantum defectscan be determined from the linear and angular coefficients in the plotof the wavenumbers of the spectral lines versus 1/n2, starting from thediffuse series where the for d orbitals (nonpenetrating orbit) is nearlyequal to zero. Then, the energy terms Tnl = R/(n l )2, where R is theRydberg constant, can be estimated by substituting the appropriatevalues of and n into this expression.

Literature Cited1. Hollenberg, J. L. J. Chem. Educ. 1966, 43, 216.2. Companion, A.; Schug, K. J. Chem. Educ. 1966, 43, 591.3. Douglas, J.; Nagy-Felsobuki, E. I. J. Chem. Educ. 1987, 64, 552.4. Herzberg, G. Atomic Spectra and Atomic Structure; Dover: Mineola,

NY, 1944.5. White, H. E. Introduction to Atomic Spectra; McGraw-Hill: New

York, 1934.6. Karplus, M.; Porter, R. N. Atoms and Molecules; W. A. Benjamin:

New York, 1970.7. Slater, J. C. Phys. Rev. 1930, 36, 57.8. Clementi, E.; Raimondi, D. L. J. Chem. Phys. 1963, 38, 2686.9. Brink, C. P. J. Chem. Educ. 1991, 68, 376.

10. Miller, K. J. J. Chem. Educ. 1974, 51, 805.11. Moore, C. E. Atomic Energy Levels, Vol. 1; U.S. National Bureau

of Standards: Washington, DC, 1949.12. Stafford, F. E.; Wortman, J. H. J. Chem. Educ. 1962, 39, 630.13. McSwiney, H. D.; Peters, D. W.; Griffith, W. B. Jr.; Mathews, C. W.

J. Chem. Educ. 1989, 66, 857.

seulaVerutaretiLdnasmreTygrenEdetaluclaC.2elbaTmotAmuidoSehtrof

nTns mc/

1 Tnp mc/1 Tnd mc/

1

dclac fer 11 )%( a dclac fer 11 )%( a dclac fer 11 )%( a

3 764,14 054,14 40.0 344,42 674,42 31.0 582,21 772,21 60.0

4 376,51 017,51 32.0 842,11 771,11 36.0 019,6 109,6 51.0

5 072,8 942,8 52.0 754,6 704,6 87.0 224,4 314,4 02.0

6 970,5 770,5 40.0 502,4 251,4 72.1 170,3 260,3 92.0

7 734,3 834,3 30.0 569,2 809,2 79.1 652,2 942,2 13.0

8 684,2 184,2 02.0 802,2 151,2 56.2 727,1 127,1 14.0

9 588,1 578,1 15.0 117,1 556,1 83.3 563,1 953,1 44.0

01 084,1 764,1 29.0 763,1 213,1 91.4 601,1 001,1 45.0aRelative error.

Z*nd(Z*nd)2Z*np(Z*np)2

dnasegrahCraelcuNevitceffEdetaluclaC.1elbaTnortcelEecnelaVmuidoSehtrofsrotcaFgnidleihS

nseireSprahS seireSlapicnirP seireSesuffiD

ns np

3 5573.3 48.1 61.9 8989.1 14.1 95.9 0000.1 00.1

4 1862.2 15.1 94.9 1726.1 72.1 37.9 0000.1 00.1

5 9968.1 73.1 36.9 1064.1 12.1 97.9 0000.1 00.1

6 7356.1 82.1 27.9 1963.1 71.1 38.9 0000.1 00.1

7 3325.1 32.1 77.9 2413.1 51.1 58.9 0000.1 00.1

8 7834.1 71.1 38.9 6872.1 31.1 78.9 0000.1 00.1

9 6083.1 61.1 48.9 1452.1 21.1 88.9 0000.1 00.1

01 1933.1 41.1 68.9 7632.1 11.1 98.9 0000.1 00.1

(Z*ns)2 Z*ns